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\subsection{Neutrino Oscillation}  The motivation for neutrino oscillation was discused in section \ref{sect:osc-motiv}. To see how a neutrino acutally "oscillates," lets assume, that we have a neutrino that intially has definite flavor. That is  \begin{equation}  \mid \nu (0) \rangle = A \mid \nu_{\alpha} \rangle  \end{equation}  Where $\alpha$ ranges over (e, $\mu$, and $\tau$) and stand for the respective flavor state.  Since $\mid \nu_{\alpha} \rangle$ are eigenstates (OF WHAT?), $\mid \nu(t) \rangle$ is given in natural units by  \begin{equation}  \mid \nu (x, t) \rangle = A e^{- i (E_{\alpha} t - \vec{p_{\alpha}} \cdot \vec{x})} \mid \nu_{\alpha} (0) \rangle  \end{equation}  To find the energy of the neutrino, we use the fact that they are traveling close to the speed of light. In this limit, $p_{\alpha} \gg m_{\alpha}$ and we can approximate the energy of the neutrinos using the relativistic equation  \begin{equation}  \begin{split}  E_{\alpha}^2 &= p_{\alpha}^2 + m_{\alpha}^2 \\  &= p_{\alpha}^2 (1 + \frac{m_{\alpha}^2}{p_{\alpha}^2}) \\  E_{\alpha} &= p_{\alpha} \sqrt{(1 + \frac{m_{\alpha}^2}{p_{\alpha}^2})} \\  &\approx p_{\alpha} (1 + \frac{m_{\alpha}^2}{2 p_{\alpha}^2}) \\  &= p_{\alpha} + \frac{m_{\alpha}^2}{2 p_{\alpha}}  \end{split}  \end{equation}  Since $E^2 = p^2 + m^2$ and $p_{\alpha} \gg m_{\alpha}$,   \begin{equation}  \begin{split}  p_{\alpha} &\approx E \\  \implies E_{\alpha} &\approx p_{\alpha} + \frac{m_{\alpha}^2}{2 E}  \end{split}  \end{equation}  Plugging this into our expression for $\mid \nu_{\alpha}(x, t) \rangle$  \begin{equation}  \begin{split}  \mid \nu_{\alpha} (x, t) \rangle &= A e^{- i (E_{\alpha} t - \vec{p_{\alpha}} \cdot \vec{x})} \mid \nu_{\alpha} (0) \rangle \\  &= A e^{-i(p_{\alpha} + \frac{m_{\alpha}^2}{2 E})t} e^{- \vec{p_{\alpha}} \cdot \vec{x}} \mid \nu_{\alpha}(0) \rangle \\  &= A e^{- i E_{\alpha} t}e^{i (E_{\alpha} - \frac{m_{\alpha}^2}{2 E}) \cdot \vec{x})} \mid \nu_{\alpha}(0) \rangle \\   &\approx A e^{- i E_{\alpha} t}e^{ -i \frac{m_{\alpha}^2}{2 E} \cdot \vec{x}} \mid \nu_{\alpha}(0) \rangle \\   \end{split}  \end{equation}  Ignoring the time-dependent phase component, the wavefunction for a neutrino as it travels down the particle accelerator is  \begin{equation}  \mid\nu_{\alpha}(x)\rangle = A e^{-i m_{i}^2 x /2E} \mid \nu_{\alpha}(0) \rangle  \end{equation}  Therefore, the probability to transition from one flavor state to another is given by  \begin{equation}  \begin{split}  P(\nu_{\alpha} \rightarrow \nu_{\beta}) & = \mid \langle \nu_{\alpha} \mid \nu_{\beta} (t) \rangle \mid ^2 \\  & = \mid A_{\alpha}^{*}A_{\beta} e^{-i \cdot m_{\alpha}^2 t /2E} \mid^2 \\  & \equiv \mid U_{\alpha \beta} e^{-i \cdot m_{\alpha}^2 t /2E} \mid^2  \end{split}  \end{equation}  Where $U_{\alpha \beta}$ is a unitary matrix called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix \cite{Kopp_2006}. As of 2012, the values for this matrix have been experimentally measured to be \cite{Fogli_2012}:  \begin{equation}  U_{\alpha \beta} = \begin{bmatrix}  0.82 & 0.54 & -0.15 \\  -0.35 & 0.70 & 0.61 \\  0.44 & -0.45 & 0.77  \end{bmatrix}  \end{equation}